Teaching Linear versus Nonlinear Functions through Discovery

A few years ago when I was co-teaching with one of my colleagues I wanted to try out the concept of discovery learning. I saw something about it in the resources we were using. So, I thought we could give it a try.

We were going to be teaching the difference between linear and nonlinear functions. It seems like a really simple concept. You just look at the graph and if it’s a straight line, then it’s a linear function. And if it’s not a straight line, then it’s a nonlinear function. But, as all teachers know, nothing is that simple. You don’t just tell kids stuff and they remember it. You can’t just talk about something for 10 minutes and then kids remember forever. If it were that simple, then we really wouldn’t need teachers.

The birth of the Linear vs. Nonlinear Discovery Lab

As my co teacher and I looked at this topic again, we decided that it was totally “gettable” by all students. And so we started plotting how to incorporate what we’d learned about discovery learning. We created and implemented a discovery learning opportunity for students that would help all of them understand what distinguishes linear functions from nonlinear functions. This experience was such a hit, that I soon became obsessed with creating more discovery learning opportunities for students on other math concepts, opportunities I began calling discovery labs (read more about the genesis of discovery labs here).

This discovery lab is still one of my very favorite activities to do with students. Today I’ll share with you how I use guiding questions and online tools to help students create their own understanding about linear & nonlinear functions. And I’ll let you know how you can get all the materials you need to teach this whole lesson for free.

Introduction to the discovery lab

So, the other day when I did this discovery lab, I asked the students the following question as an anticipatory set:

What do you know about functions? How are they related to graphs, tables, and equations?

Well, I got a lot of blank stares and a “Conjunction, junction, what’s your function?” from the back of the room. I didn’t realize that they didn’t know the word “functions”. This didn’t derail my lesson because they don’t need to know the word to make the lab work. It just meant that I needed to make sure that they attached this word to linear equations in all forms. So, I gave them a brief mini-lesson showing them examples of functions. It turned out that they recognized functions, they just weren’t familiar with that word to talk about them.

I then explained to them the when we saw the word “linear” that meant a function made a straight line. “Nonlinear” wouldn’t be a straight line. They added that note to the top of their discovery lab as a reference throughout the activity. With this new vocabulary under their belts, it was now time to turn the class over to the students to get to discovering.

Modeling a Trial

From my previous few discovery labs working with my advanced classes, I learned that I needed to give them a little more direction and support. Just because they’re advanced doesn’t mean they know how to do everything. So, I walked them through the first trial they were asked to conduct in this discovery lab so they could clearly see what they were expected to do.

I gave students a fill-in the blank sentence of how I wanted them to write their prediction. Together, we came up with our first prediction and recorded it under #1. Then, we moved on to doing the rest of Trial #1 together.

In each trial, students had to do four things:

Make a prediction.

Input the equation into the graphing calculator.

Sketch the line it forms.

Identify if it’s a linear or nonlinear function.

Next, I showed them how to enter the function y=2x + 1 into the Desmos graphing calculator. We used the Desmos calculator on an iPad. Students shared one iPad per partnership. You could also use the Desmos graphing calculator online with any computer.

I typed the equation into the calculator and voila, the line of that function showed up. The function in Trial #1 is a straight line, so we sketched it together. I didn’t emphasize copying the line perfectly because that’s not the focus of the lesson.

Finally, I modeled the sentence I wanted them to write at the end of the trial in response to the question, “Is the function linear or nonlinear? How do you know?” I didn’t require students to write the same exact sentence that I wrote, but it was good as a support for some kids. One of my classes is almost half ELL students and it was especially helpful for them. After modeling, the responses I got for the predictions and the conclusions were definitely better than in previous years before just from taking a bit of time to give students a model.

After doing the first trial as a class, students worked with their partners to do the next nine trials together, looking at a variety of functions.

Students perform trials and make observations

At this point students chose a partner and were off on their exploration of linear and nonlinear functions for the next 20-25 minutes. I honestly think this lesson is my favorite lesson of the whole year. The conversations that students were having were amazing. They were so engaged and excited. So many of them wanted to tell me what they were noticing. There was real shock on their faces when a parabola or an asymptote would appear.

They were nerdy excited to discover what would happen with the equations when they were put on a graph. They had a lot of interesting questions as the trials happened.

Student make their own equations

The last two trials don’t have equations. The students come up with their own equations. This gives them the opportunity to play around with the idea of linear and nonlinear. Most, but not all of the students got to this part. It’s like a bonus part. My students were pretty excited when they got here and they tried different things. They tried really big numbers, played around with the exponents, and sometimes they didn’t have an x in their equation at all. This gave students a chance to experiment and extended the discovery in this activity.

Conclusions about the trials

The whole idea of a discovery lab is for students to start creating their own understanding of a concept. When we get to the conclusions we get to see if they have started this process or not. Some students have created a great foundation to build on as the unit progresses. And for those who still feel a little lost, they at least have a better starting place to eventually “get it” and really understand the differences between linear and nonlinear functions.

Most of my students drew the basic conclusion that if there was a variable on the x, then the line wouldn’t be linear. The class period that we spent on this discovery served as a background building activity. As a group, my students definitely drew the conclusion that linear functions create straight lines and that nonlinear functions don’t form straight lines. Basically, after the discovery lab everyone knew more about linear functions (and what nonlinear functions are), but they still didn’t understand the whole concept perfectly.

Giving students the opportunity to express their understanding at the end of the discovery activity makes them accountable for it. It’s very powerful because they have to articulate the concept instead of relying on the teacher’s interpretation.

Questions they have about linear and nonlinear functions

At the very end I asked students what questions they have about this topic. One thing I realized reviewing their answers is that next year I need to help them write questions that show what they wonder about after this activity. My students assumed it was supposed to be a question of something that they found confusing, but really I was looking for more of a question of wondering.

Here are some of the questions students had:

-What is the vocabulary for linear and nonlinear?

-Can there be a line that is straight, but has one part that is not straight?

-What happens to the exponent when it is not at the end of the equation?

-I don’t really have any questions, but I would like to learn more of how to do the graphs. What is a parabola and an asymptote?

-Is there an easy way to predict if the line will be linear or nonlinear?

Reading their questions gave me a lot of insight into what they’re thinking and what to emphasize when we did notes. I actually let them try to fill-in the notes first before I told them the answers. That worked really well. They had such a strong background after the discovery lab that I didn’t have to do a lot of explaining at all.

Taking it to your classroom

If you want to try a discovery lab and you teach this topic, this Linear and Nonlinear Discovery Lab Activity works great as a first try. You’ll be amazed at how engaged students are in creating their own understanding.

Pro tip: Don’t forget to make sure you guide them enough to be successful, but not too much to where you are a crutch. Let kids struggle, and think, and work for the understanding. If you do that, then students will remember what they’re learning because they’ll have a place in their brain to connect it to.